Miscible Displacement In Porous Media Research Articles

Nondilute effects reshape miscible displacement in porous media

We provide an experimental benchmark for modeling nondilute miscible mixing in porous media, which emerges in CO${}_{2}$ sequestration and solvent-based enhanced oil recovery. In a near-fracture porous medium, we show that conventional Darcy-Fickian coupling fails to predict nondilute fluid-fluid displacement, although it works perfectly for dilute systems. Nondilute fluid systems show ``diffusive'' mixing kinetics, wedge-like mixing front, and significant convection at the front that is perpendicular to the concentration gradient. These above abnormal phenomena are rationalized by Korteweg stress, which is still challenging in modeling.

Competition of gravity and viscous forces in miscible vertical displacement in a three-dimensional porous medium

When viscosity and density contrast exist in the vertical miscible displacement in porous media between two fluids, the interplay between the viscous force and gravity determines the interface stability. Two stability criteria are derived to determine the interface stability. Hill's and Dumore's stability criteria are used to determine the interface stability of the sharp and diffused interface, respectively. In this study, we visualized the crossover between unstable displacement and stable displacement for a vertical displacement in porous media using microfocused x-ray computed tomography. The experiments were divided into four possible configurations: (1) unconditionally stable (gravitationally stable-viscously stable), (2) unconditionally unstable (gravitationally unstable-viscously unstable), (3) conditionally stable (gravitationally stable-viscously unstable), and (4) conditionally stable (gravitationally unstable, viscously stable). The structure of the displacement interface was visualized for the critical velocity ratio (V/Vc) in the range of 0.5–11.9. In the conditionally stable configurations 3 and 4, a crossover between stable and unstable displacements was observed. We found that Dumore's stability criterion is more appropriate for predicting interface stability than Hill's stability criterion. Viscous fingering occurs in configuration 3 when V/Vc is higher than Dumore's critical velocity, whereas gravity fingering occurs in configuration 4 when V/Vc is lower than Dumore's critical velocity. Similar events in two-dimensional experiments, such as tip-splitting, shielding, and coalescence, were also observed three-dimensionally. The significant changes in the mixing length and sweep efficiency signify the crossover between the stable and unstable displacements.

Bound-preserving discontinuous Galerkin methods with second-order implicit pressure explicit concentration time marching for compressible miscible displacements in porous media

In this paper, we construct bound-preserving interior penalty discontinuous Galerkin (IPDG) methods with a second-order implicit pressure explicit concentration (SIPEC) time marching for the coupled system of two-component compressible miscible displacements. The SIPEC method is a crucial innovation based on the traditional second-order strong-stability-preserving Runge-Kutta (SSP-RK2) method. The main idea is to treat the pressure equation implicitly and the concentration equation explicitly. However, this treatment would result in a first-order accurate scheme. Therefore, in all previous works, only the combination of forward and backward Euler time integrations was considered. In this paper, we propose a correction stage to compensate for the second-order accuracy in each time step. There are two main difficulties in constructing a second-order scheme. Firstly, in the concentration equation, correction of the diffusion term will cause anti-diffusion, leading to malfunction of the bound-preserving technique. We can deal with the velocity in the diffusion term explicitly to avoid correction of the diffusion term. Secondly, we need to ensure that the bound-preserving technique for the convection and source terms can be applied when the correction stage has been established. In fact, in the correction stage, the new approximation to the concentration can be chosen as the numerical solution in the previous stage, so the numerical cell averages are positivity-preserving. Moreover, we use the same correction technique for the pressure, so that the consistent flux pairs would guarantee the preservation of the upper bound 1 of the concentration. Numerical experiments will be given to demonstrate that the proposed scheme can reduce the computational cost significantly compared with explicit schemes if the diffusion coefficient D is small in the concentration equation. Moreover, the proposed method also yields much larger cfl number compared with first-order implicit pressure explicit concentration schemes. Moreover, the effectiveness of the bound-preserving technique will also be presented.

New analysis and recovery technique of mixed FEMs for compressible miscible displacement in porous media

New analysis and recovery technique of mixed FEMs for compressible miscible displacement in porous media

An Lpspaces-based formulation yielding a new fully mixed finite element method for the coupled Darcy and heat equations

AbstractIn this work we present and analyse a new fully mixed finite element method for the nonlinear problem given by the coupling of the Darcy and heat equations. Besides the velocity, pressure and temperature variables of the fluid, our approach is based on the introduction of the pseudoheat flux as a further unknown. As a consequence of it, and due to the convective term involving the velocity and the temperature, we arrive at saddle point-type schemes in Banach spaces for both equations. In particular, and as suggested by the solvability of a related Neumann problem to be employed in the analysis, we need to make convenient choices of the Lebesgue and ${\textrm {H}}(div)$-type spaces to which the unknowns and test functions belong. The resulting coupled formulation is then written equivalently as a fixed-point operator, so that the classical Banach theorem, combined with the corresponding Babuška–Brezzi theory, the Banach–Nečas–Babuška theorem, suitable operators mapping Lebesgue spaces into themselves, regularity assumptions and the aforementioned Neumann problem, are employed to establish the unique solvability of the continuous formulation. Under standard hypotheses satisfied by generic finite element subspaces, the associated Galerkin scheme is analysed similarly and the Brouwer theorem yields existence of a solution. The respective a priori error analysis is also derived. Then, Raviart–Thomas elements of order $k\ge 0$ for the pseudoheat and the velocity and discontinuous piecewise polynomials of degree $\le k$ for the pressure and the temperature are shown to satisfy those hypotheses in the two-dimensional case. Several numerical examples illustrating the performance and convergence of the method are reported, including an application into the equivalent problem of miscible displacement in porous media.

Conservative numerical methods for the reinterpreted discrete fracture model on non-conforming meshes and their applications in contaminant transportation in fractured porous media

The discrete fracture model (DFM) has been widely used to simulate fluid flow in fractured porous media. Traditional DFM is considered to be limited on conforming meshes, hence significant difficulty may arise in generating high-quality unstructured meshes due to the complexity of the fracture networks. Recently, Xu and Yang reinterpreted DFM and demonstrated that it can actually be extended to non-conforming meshes without any essential changes. However, the continuous Galerkin (CG) method was applied and the local mass conservation was missing. This paper is a follow-up work, and we apply the interior penalty discontinuous Galerkin (IPDG) method and enriched Galerkin (EG) method for the pressure equation. With the numerical fluxes, the local mass is conservative. As an application, we combine the reinterpreted DFM (RDFM) with the incompressible miscible displacements in porous media. The bound-preserving techniques are applied to the coupled system. We can theoretically guarantee that the concentration is between 0 and 1. Finally, several numerical experiments are given to demonstrate the good performance of the RDFM based on the above two methods on non-conforming meshes and the effectiveness of the bound-preserving technique.

Experimental Observation of Two Distinct Finger Regimes During Miscible Displacement in Fracture

Miscible displacement of two-phase fluids in rough fractures is relevant to some industrial processes, including enhanced oil recovery and geological carbon sequestration. When a less viscous fluid displaces another more viscous fluid, finger instability occurs. Previous works focused on miscible displacement in porous media or Hele-Shaw, but the experimental study was rarely reported for rough fractures. Here, we perform visualization experiments of water displacing glycerol in a transparent fracture model to investigate the effects of flow rate and diffusion in miscible displacement. We quantify the displacement patterns using the sweep efficiency, the mixing length, and the relative contact area. We observe two distinct displacement regimes: dominant finger regime and multiple fingers regime. A critical Peclet number Pe is obtained to identify such two regimes. Below the critical Pe, the channel forms, and the displacement is the dominant finger regime, which results in low sweep efficiency and linearly growth of mixing length at late time. Above this critical Pe, intensive tip-splitting events result in the formation of dendritic displacement pattern, and the displacement is multiple fingers regime, slowing down the growth rate of mixing length at late time and contributes to the higher sweep efficiency. Our work shows a critical Pe that separates the two distinct regimes and improves our understanding of the evolution of the miscible displacement fronts in rough fractures.

A Pore-Scale Model for Dispersion and Mass Transfer during Acoustically Assisted Miscible Displacements in Porous Media

Enhancing flow and transport in porous media by acoustic stimulation has been investigated for the past several decades. Most studies focused on the effect of sound propagation in immiscible displa...

English

We demonstrate some a priori estimates of a scheme using stabilization and hybrid interfaces applying to partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection-diffusion-dispersion equation on the concentration of invading fluid. The anisotropic diffusion operators in both equations require special care while discretizing by a finite volume method SUSHI. Later, we present some numerical experiments.

Hp-discontinuous Galerkin Finite Element Method for Incompressible Miscible Displacement in Porous Media

We used (hp-DGFEM) for solving the coupled system of incompressible miscible displacement problem in porous media. Flow and transport equations are solved by hp-symmetric interior penalty Discontinuous Galerkin (hp-SIPG) method. We prove the error analysis in the energy norm to the coupled system of incompressible one-phase flow problem.

Modeling and simulation of partially miscible two-phase flow with kinetics mass transfer

Multiphase flow equations for two components (for instance water and hydrogen) and two phases (liquid and gas), with equilibrium phase exchange, have been used to simulate the process of miscible displacement in porous media. Laboratory and field studies have shown that this assumption fails under certain circumstances especially for accurate description of the pollution and a finite transfer velocity, called here the kinetics. We propose a numerical scheme based on a two-step convection/diffusion-relaxation strategy to simulate the non-equilibrium model. In the first step we solve the intra-phase transfer (convection/diffusion) working with liquid saturation, liquid pressure and dissolved hydrogen concentration as primary variables. In the second step, we address the interphase transfer using the solubility relation that is solved by projection on the equilibrium state. This technique also ensures the positivity for the liquid saturation and produces energy estimates. One important advantage of this approach is the fact that the simulation can be easily adapted to different linear and non-linear equilibrium laws between phases. Another advantage is that our proposed method includes the kinetics in a projection step and keeps the displacement of the components. We implemented this new model in in-house simulation code and present numerical results comparing the model with equilibrium and non-equilibrium one.

Numerical modeling of viscous fingering during miscible displacement of oil by a paraffinic solvent in the presence of asphaltene precipitation and deposition

We study the mixing dynamics of a paraffinic solvent with oil in the context of miscible displacement in porous media for enhanced oil recovery. Typically, injected solvents are less viscous compared to the resident oil, leading to the rise of interfacial instabilities known as viscous fingering. This, in turn, accelerates precipitation of initially dissolved asphaltene particles in the oil. The formed precipitates are carried with the flow and may deposit in the porous media, altering porosity and permeability fields. Although asphaltene precipitation and deposition are inevitable in most of the solvent-based recovery techniques, they have been neglected in the previous works of miscible displacement due to their added complexity. In this work, we present a rigorous formulation of the physics of the problem and conduct high-resolution nonlinear numerical simulations to investigate the effects of asphaltene precipitation and deposition and the resulting formation damage. For this purpose, the related governing equations are derived based on the velocity-vorticity formulation, and a high-order hybrid scheme is applied to solve them. This type of handling enables us to simulate viscous fingering at large viscosity ratios and Peclet numbers, beyond what has been previously reported in the literature. Furthermore, experimental measurements are conducted for viscosity and asphaltene content of the oil at various solvent concentrations and are used in the simulations. The results of our numerical simulations reveal that despite having a great impact on the permeability field, asphaltene precipitation and deposition do not influence the growth of viscous fingers, noticeably, which is due to the substantial viscosity ratio of the studied system. However, deposition of the formed precipitates leads to in-situ upgrading of the oil and thus enhances the quality of the produced liquid. Finally, regarding the effect of the choice of viscosity mixing rule, we compare two commonly used viscosity mixing rules in the previous works (exponential and quarter-power) and elaborate on their main differences in terms of the observed dynamics.

Convergence Analysis of Crank–Nicolson Galerkin–Galerkin FEMs for Miscible Displacement in Porous Media

We propose a fully discrete linearized Crank–Nicolson Galerkin–Galerkin finite element method for solving the partial differential equations which govern incompressible miscible flow in porous media. We prove optimal-order convergence of the fully discrete finite element solutions without any restrictions on the step size of time discretization. Numerical examples are provided to illustrate the theoretical results.

Numerical analysis of a SUSHI scheme for an elliptic-parabolic system modeling miscible fluid flows in porous media

In this paper, we prove the convergence of a schema using stabilisation and hybrid interfaces of partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations:an anisotropic diffusion equation on the pressure and a convection-diffusion dispersion equation on the concentration of invading fluid. The anisotropicdiffusion operators in both equations require special care while discretizing bya finite volume method SUSHI. Later, we present some numerical experiments.

High-order bound-preserving finite difference methods for miscible displacements in porous media

In this paper, we develop high-order bound-preserving (BP) finite difference (FD) methods for the coupled system of compressible miscible displacements. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the jth component, cj, should be between 0 and 1. It is well known that cj does not satisfy a maximum-principle. Hence most of the existing BP techniques cannot be applied directly. The main idea in this paper is to construct the positivity-preserving techniques to all cj′s and enforce ∑jcj=1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation and choose suitable “consistent” numerical fluxes in the pressure and concentration equations. Recently, the high-order BP discontinuous Galerkin (DG) methods for miscible displacements were introduced in [4]. However, the BP technique for DG methods is not straightforward extendable to high-order FD schemes. There are two main difficulties. Firstly, it is not easy to determine the time step size in the BP technique. In finite difference schemes, we need to choose suitable time step size first and then apply the flux limiter to the numerical fluxes. Subsequently, we can compute the source term in the concentration equation, leading to a new time step constraint that may not be satisfied by the time step size applied in the flux limiter. Therefore, it would be very difficult to determine how large the time step is. Secondly, the general treatment for the diffusion term, e.g. centered difference, in miscible displacements may require a stencil whose size is larger than that for the convection term. It would be better to construct a new spatial discretization for the diffusion term such that a smaller stencil can be used. In this paper, we will solve both problems. We first construct a special discretization of the convection term, which yields the desired approximations of the source. Then we can find out the time step size that suitable for the BP technique and apply the flux limiters. Moreover, we will also construct a special algorithm for the diffusion term whose stencil is the same as that used for the convection term. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.

Pore-scale modeling of coupled thermal and solutal dispersion in double diffusive-advective flows through porous media

In double diffusive-advective flows through porous media, the combined effects of fluid and solid properties, pore space geometry, and flow conditions on the transport rates and distribution of the thermal and solutal fronts can be characterized by dispersion coefficients. In this study, conjugate heat and solute transport in the process of miscible displacement through porous media is studied through pore-scale numerical simulations in 2D and 3D digital representations of the unconsolidated sandpacks. The temperature and solute concentration distributions during both viscously stable and unstable displacements are obtained by solving the point equations of flow and transport on the discretized computational domains. Next, the longitudinal components of the solute and thermal dispersion coefficients are determined by fitting the effluent profiles to the analytical solutions of the advection-diffusion equations for the transport of mass and heat, respectively. The length scales of the solutal and thermal transition zones are also measured along the principal direction of the flow and then correlated with the displacement time. In stable displacements, different dispersion regimes are identified based on the magnitude of the solutal and thermal Peclet numbers. It is observed that the onset of convective spreading is different in the 2D and 3D simulation runs. In unstable displacements, the results indicate that solutal dispersion is more affected by thermo-solutal viscous fingering compared to thermal dispersion. Moreover, as the viscosity contrast across the thermal and solutal fronts increases, the dynamics of the mixing length growth gradually changes from dispersion-dominated toward fingering-dominated. Finally, the results show that two-dimensional simulation runs may not completely characterize the dispersion-related phenomena that occurs during heat and solute transfer in a real porous medium. Therefore, it is essential to conduct three-dimensional runs to capture all the underlying mechanisms contributing to the mixing and dispersion in porous media. The outcomes of this study can pave the way for an optimal design and implementation of an efficient non-isothermal miscible displacement in porous media.

The error analysis of linearized discontinuous Galerkin finite element method for incompressible miscible displacement in porous media

We present a linearization scheme for discontinuous Galerkin method for solving the fully coupled one-phase flow problem. The pressure and concentration equations are solved by symmetric interior penalty Discontinuous Galerkin (SIPG) method and we discretized the model using (DGM) space and backward Euler in time. We prove the error estimate in the energy norm.

New Analysis of Galerkin FEMs for Miscible Displacement in Porous Media

The paper is concerned with optimal error estimates of classical Galerkin FEMs for the equations of incompressible miscible flows in porous media. The analysis done in the last several decades shows that classical Galerkin FEMs provide the numerical concentration of the accuracy $$O(\tau ^k+h^{r+1} + h^s)$$ in $$L^2$$ -norm. This analysis leads to the use of higher order finite element approximation to the pressure than that to the concentration in various numerical simulations to achieve the best rate of convergence. However, this error estimate is not optimal. The purpose of this paper is to establish the optimal $$L^2$$ error estimate $$O(\tau ^k + h^{r+1} + h^{s+1})$$ , from which one can see that the best convergence rate can be achieved by taking the same order $$(r=s$$ ) approximation to the concentration and pressure. Clearly Galerkin FEMs with $$r=s$$ are less expensive in computation and easier for implementation. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis.

Analysis of fully discrete FEM for miscible displacement in porous media with Bear–Scheidegger diffusion tensor

Fully discrete Galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly-used Bear--Scheidegger diffusion-dispersion tensor: $$ D({\bf u}) = \gamma d_m I + |{\bf u}|\bigg( \alpha_T I + (\alpha_L - \alpha_T) \frac{{\bf u} \otimes {\bf u}}{|{\bf u}|^2}\bigg) \, . $$ Previous works on optimal-order $L^\infty(0,T;L^2)$-norm error estimate required the regularity assumption $\nabla_x\partial_tD({\bf u}(x,t)) \in L^\infty(0,T;L^\infty(\Omega))$, while the Bear--Scheidegger diffusion-dispersion tensor is only Lipschitz continuous even for a smooth velocity field ${\bf u}$. In terms of the maximal $L^p$-regularity of fully discrete finite element solutions of parabolic equations, optimal error estimate in $L^p(0,T;L^q)$-norm and almost optimal error estimate in $L^\infty(0,T;L^q)$-norm are established under the assumption of $D({\bf u})$ being Lipschitz continuous with respect to ${\bf u}$.

Dynamics of Miscible Nanocatalytic Reactive Flows in Porous Media

Catalysis at the nanoscale plays an increasingly important role in subjects ranging from energy and environment to medical sciences, but our physical understanding of the dynamics and effectiveness of nanocatalysts in porous media is lacking. The authors theoretically investigate the factors needed for success in $i\phantom{\rule{0}{0ex}}n$ $s\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}u$ nanocatalyst applications, such as heavy-oil upgrading and soil remediation, and they discuss the physical mechanisms behind the dynamics of mixing during nanocatalytic miscible displacement in porous media. This insight will aid the optimal design and utilization of nanocatalysts.